Point-Defect Optical Transitions and Thermal Ionization Energies
from Quantum Monte Carlo Methods: Application to F-center Defect in MgO
Elif Ertekin,∗ Lucas K. Wagner,† and Jeffrey C. Grossman‡
We present an approach to calculation of point defect optical and thermal ionization energies based
on the highly accurate quantum Monte Carlo methods. The use of an inherently many–body
theory that directly treats electron correlation offers many improvements over the typically-employed
density functional theory Kohn-Sham description. In particular, the use of quantum Monte Carlo
methods can help overcome the band gap problem and obviate the need for ad-hoc corrections. We
demonstrate our approach to the calculation of the optical and thermal ionization energies of the
F-center defect in magnesium oxide, and obtain excellent agreement with experimental and/or other
high-accuracy computational results.
From electronics to optoelectronics to photovoltaics,
point defects influence and even dominate the properties of semiconducting materials [1–6]. Quantitative descriptions of the effect of point defects on electronic,
optical, and transport properties is critical to enabling
point-defect engineering for materials design. However, accurate prediction of point-defect energetics, thermal ionization energies, and optical transition energies
from first principles remains a challenge. Currently, the
most widely-used approach based on conventional density
functional theory (DFT) suffers from poor descriptions of
band gaps that render difficult the accurate description of
mid-gap defect states [5, 7–9]. Here we demonstrate that,
by contrast, an inherently many-body approach based on
quantum Monte Carlo (QMC) methods [10, 11] can eliminate these problems and enable high-accuracy calculations of point defect optical and thermal ionization energies. Our computed optical transition energies are in excellent agreement with experimental and/or other highaccuracy computational results for the same system [12],
and demonstrate that QMC can obtain quantitatively
accurate descriptions.
QMC methods comprise a suite of stochastic tools that
enable calculations of material properties based on the
many-particle Schrödinger equation. Because of their
direct treatment of electron correlation, QMC methods
are among the most accurate electronic structure approaches available today, and demonstrate a long and
distinguished record of ground-breaking and benchmarking calculations [10, 11, 13]. In comparison to the other
“beyond-DFT” techniques that are currently explored for
calculation of point defect properties (DFT+U, hybrid
DFT, and the GW method), QMC is directly based on
the true many-body Schrödinger equation and offers the
possibility of parameter-free accurate band gaps and total energies. The application of QMC techniques to point
defects in solids is still a relatively new field. To date, a
handful of studies have been carried out to compute defect formation energies: interstitials in silicon [14–16], vacancies in diamond [15], the Schottky defect in MgO [17],
and vacancies and interstitials in aluminum [18].
In this letter, we illustrate the application of the QMC
method to the F-center defect in magnesium oxide (MgO)
by computing defect formation energies, thermal ionization levels, and optical ionization energies (although the
approach is general and can be extended to other materials of interest). The F-center defect (oxygen vacancy)
in MgO is a typical example of an intrinsic point defect in a binary ionic compound [22–27]. Despite its apparent simplicity, its properties as deduced from optical
absorption and luminescence studies have proven somewhat ambiguous. Experimental characterization of the
F-center in its neutral (F0 ) and singly ionized (F+1 ) state
has been complicated by their nearly identical optical absorption energies [25–27]. These energies have been corroborated by recent GW calculations [12]; however, these
calculations also predicted optical emission energies that
8
4
Energy (eV)
arXiv:1210.0940v1 [cond-mat.mtrl-sci] 2 Oct 2012
Dept. of Materials Science & Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
(Dated: October 4, 2012)
t1u
t1u
0
a1g
-4
a1g
-8
K
Γ
X
W L
Γ
FIG. 1.
(Color online). Left: The electronic band structure of a 64 atom MgO supercell containing a single oxygen
vacancy, calculated within DFT-PBE. The neutral oxygen
vacancy introduces a localized mid gap defect level of symmetry a1g . There is also a triply degenerate excited defect
level in the conduction band of t1u symmetry. Right: The
corresponding a1g and t1u Kohn-Sham states plotted at the
Γ-point, showing the localized nature in the vicinity of the
vacancy.
2
Lattice const (Å)
Coh. En. (eV/MgO)
OP (eV)
QP (eV)
IP (eV)
EA (eV)
DFT-PBE
DMC
Exp [19–21]
4.25
9.50
4.83
–
–
–
4.22
10.18(5)
7.96(6)
7.9(1)
3.28(7)
11.17(7)
4.216
10.5
7.78
7.84
–
–
TABLE I. Comparison of lattice constant, atomization energy, and band gap in MgO solid according to DFT, DMC,
and in experiment. All computed results are obtained using
Ne-core pseudo potentials for magnesium. The optical band
gap is determined in DFT from the Kohn-Sham levels, and in
DMC from the extrapolated optical excitation energy. Error
bars are shown in parenthesis. Note that the IP and EA are
given with respect to the average potential in the supercell.
are substantially different from the assigned experimental
values, causing the authors to suggest a reinvestigation
of the experimental observations. A particularly compelling possibility is to explore the F-center defect using
distinct high-accuracy first-principles techniques to compare results. Our results - calculated independently using
QMC methods - corroborate the GW results and further
invite reassessment of the experimental data for the optical emission.
We first compute the properties of the F-center defect in MgO within a DFT [28, 29] framework as implemented in the SIESTA code [30], employing the PerdewBurke-Ernzerhof [31] approximation to the exchange correlation potential. The inner core electrons are represented by Troullier-Martins pseudopotentials (leaving
the Mg 3s and O 2s, 2p electrons in valence), and the
Kohn-Sham orbitals are represented by a linear combination of numerical pseudo atomic orbitals expanded in
a triple-ζ with polarization Gaussian basis set. For bulk
rocksalt MgO, in agreement with previous DFT calculations [32, 33] we find a lattice parameter of 4.25 Å
(4.22 Å in experiment [19]), an atomization energy of
9.50 eV/MgO (10.50 eV/MgO in experiment [19]), and
a direct band gap of 4.83 eV at the Γ-point (a considerable underestimate of the experimental band gap of 7.78
eV [20]).
The DFT band gap underestimate has a severe consequence on the prediction of mid-gap defect states, defect energetics (particularly for occupied defect levels),
and defect-induced optical absorption and emission energies. Broadly, the band gap underestimate arises because in typically-employed mappings of the interacting many-body Schrödinger equation to the DFT singleparticle “effective”-potential Kohn-Sham equations, each
electron also interacts with itself (self-interaction error [34–38]). This results in an extraneous Coulomb
repulsion that overly delocalizes electronic states. The
self-interaction error, in addition to the absence of a
derivative discontinuity in the exchange-correlation potential [34–38], results in underestimated band gaps that
have plagued DFT calculations.
In Fig. 1, we show the DFT-computed electronic band
structure of a 63 atom MgO supercell containing an F0 center defect (neutral oxygen vacancy). In agreement
with previous DFT calculations [12], the F0 -center introduces a fully-occupied mid-gap defect level of a1g symmetry into the electronic band structure; higher in the
conduction band we also find a triply-degenerate excited
defect level of t1u symmetry.
The QMC calculations reported here are computed
within fixed node diffusion Monte Carlo (DMC) as implemented in the QWalk code [39], with single-determinant
Slater-Jastrow trial wave functions constructed from the
DFT orbitals, variance-minimized Jastrow coefficients,
and a time step of 0.01 au. To establish that our choice
of pseudopotentials is reasonable, we first calculated the
bond length, electron affinity, and binding energy of the
MgO molecule within DMC. We tested both Ne and He
-core pseudopotentials for the Mg atom in the molecule,
and found that (although both give good results) the
small core pseudopotential gives a somewhat better description (see supplementary information). This suggests
that including the Mg 2s and 2p electrons improves the
description slightly; however, for the solid, the computational cost of the He-core pseudopotential was prohibitive.
To test the properties of the non-defective MgO solid
in QMC, we calculate the atomization energy, the optical band gap, and the quasiparticle band gap. All energies are calculated using the extrapolation framework
described in Refs [40–42] with supercells containing 16,
32, and 64 atoms. Further details are provided in the
supplementary information. We calculate the following
states: the ground state Eg , the Γ-point optically excited state EΓ→Γ , the positively charged state E+ , and
the negatively charged state E− . The ionization potential (IP), electron affinity (EA), quasiparticle gap (QP),
and optical gap (OP) are given by
IP = Eg − E+
(1)
EA = E− − Eg
(2)
QP = EA − IP
(3)
OP = EΓ→Γ − Eg
(4)
The results are summarized in Table I, and show excellent agreement overall with the experimental values.
The slight underestimation of the atomization energy is
likely due to the Ne-core pseudopotential, since the MgO
molecule showed a similar effect (see supplementary information), while the gap calculations are close to experiment, overestimating the gap slightly.
Now we turn to the F-center defect. We again use
64 (perfect) and 63 (with F-center) atom supercells, and
compute the defect formation energy ∆ED,q according to
X
∆ED,q = (ED,q − Eperf ) −
ni µi + q(EV + EF ) (5)
i
3
Mg-Mg (Å) O-O (Å) Relaxation (eV)
perfect
F
F+
F+2
2.98
2.99
3.09
3.17
5.96
5.96
5.90
5.84
0.003
0.545
1.182
Def. Form. Energy ΔED,q (eV)
Def. Form. Energy ΔED,q (eV)
TABLE II. DFT-computed lattice relaxations for the F, F+ ,
and F+2 center. The Mg-Mg distance denotes the separation between Mg atoms that neighbor the missing O atom;
similarly the O-O distance denotes the separation between O
atoms that neighbor the missing O atom.
6
(a) DFT
q=+2
q=+1
4
q=0
2
0
1
2
3
4
Fermi Energy EF (eV)
6
(b) QMC
q=+2
q=+1
4
q=0
2
0
2
4
6
8
Fermi Energy EF (eV)
FIG. 2. (Color online). Comparison of F-center defect formation energies and thermal ionization energies in MgO computed in DFT and DMC. The domain of the Fermi energy (xaxis) is determined by the band gap of the system according
to the computational framework; clearly the DMC description of the gap is better and obviates the need for “band-gap
corrections”. In comparison to DFT, DMC modifies somewhat the absolute value of the defect formation energies, but
maintains thermal ionization levels near mid gap. (Note that
the error bars of the DMC-computed formation energies are
smaller than the line widths in (b).)
where ED,q is the (computed) total energy of the supercell containing a defect D in the charge state q, Eperf is
the (computed) total energy of the perfect supercell, and
ni is the number of atoms of species i added to (ni > 0)
or removed from (ni < 0) the supercell to create the defect [43]. Different environmental conditions are accommodated by the set of chemical potentials µi for each
element by assuming that each is in equilibrium with a
physical reservoir such as a gas or a bulk phase. EV is
the energy of the valence band maximum (the ionization
potential in DMC), and EF is the Fermi energy refer-
enced to EV so that 0 ≤ EF ≤ Eg where Eg is the band
gap.
The thermal ionization energies, which determine the
shallow or deep nature of a defect, correspond to the
Fermi energies at which the energetically most favored
charge state of the defect changes. According to our
DFT calculations, the creation of an F0 -center results
in the formation of a filled mid-gap defect level (shown
in Fig. 1). There is very little lattice relaxation that
takes place upon removal of the O, as indicated in Table II. However, when an electron is removed from the
supercell to form the F+1 -center, in DFT we find a large
lattice relaxation as the positively charged Mg ions move
outwards away from, and the negatively charged O ions
move inwards towards, the positively charged vacancy
in conjunction with a 0.55 eV drop in energy. Further
ionizing the defect into the F+2 state in DFT results in
further lattice relaxations accompanied by an energy recovery of 1.18 eV. The DFT defect formation energies
obtained from Eq. 5 are plotted in Fig. 2a, showing thermal ionization levels near the middle of the gap. These
formation energies are computed in the Mg-rich limit so
that µM g is given by the chemical potential of solid elemental magnesium (and µM g + µO = µM gO ). In Fig. 2a,
we have used the as-computed DFT band gap for MgO,
without correction schemes to artificially open the gap.
Note that we have also ignored the interaction energy
between charged defects arising from the use of periodic
boundary conditions, which would shift upwards the lines
for the formation energy of the F+1 and F+2 center (and
reduce the defect transition levels). We will ignore this
interaction for the QMC calculations as well, and compare both methods on an even footing [44].
For the QMC calculations of total energies of perfect and defected supercells (Eperf , ED,q ) and the corresponding thermal ionization levels, we use the relaxed
lattice geometries and orbitals obtained from the DFT
calculations. For computational efficiency, only the Γ
point Kohn-Sham orbitals are used, rather than twistaveraging, since in Eq. 5, the errors in the difference
(ED,q −Eperf ) largely cancel. For instance, in DFT these
errors are 0.26 eV for the F0 -center, and < 0.1 eV for
the F+1 and F+2 -center, and do not substantively alter
our results. Figure 2b shows the defect formation energies and the thermal ionization levels as computed within
QMC. The largest single difference between the DFT and
the QMC results is, of course, the domain for the Fermi
energy 0 < EF < Eg . For this material, both DFT and
QMC place the transition levels near mid-gap [45].
From Fig. 2, the overall formation energy of the neutral defect (F0 ) is higher in QMC by approximately 1.5
eV. This increase in formation energy may be largely
attributed to the fact that DFT-PBE renders the MgO
system more delocalized while QMC captures the true
ionic nature of the solid. The overly-delocalized (more
metal-like) description in DFT makes the penalty for
Def. Form. Energy ΔED,q (eV)
4
CBM
16
3.49(0.08) eV
5.02(0.08) eV
14
F
+2
em
æ
12
abs
æ
+1
Fem
+1
abs
F
10
VBM
F+1
æ
8
æ
æ
6
0
Fabs
F0-center
CBM
F0
0
em
F
3.76(0.12) eV
5.12(0.12) eV
4
æ
em
æ
0
0.1
0.2
0.3
Mg atom disp. (A)
0
0
Fabs
Fem
QMC 5.0(1) 3.8(1)
GW[8] 4.95
3.6
Exp
5.00 3.1-3.2
abs
+1
F -center
VBM
+1
+1
Fabs
Fem
5.1(1) 3.5(1)
4.92
3.4
4.95 2.3-2.4
FIG. 3.
(Color online). Optical absorption and emission
energies (in eV) computed in DMC for the F-center defect in
MgO. The DMC absorption energies are in excellent agreement with experiment and recently published GW results.
The DMC emission energies are in disagreement with the experimentally assigned values, but match closely to the GW
results.
bond breaking too small, and consequently the defect formation energy too small as well. This over-delocalization
results in the underestimated DFT band gap; the effect
on computed energetics are significant for defects with
occupied mid-gap defect levels such as the F0 center. As
a result of the underestimated band gap, in DFT the
Kohn-Sham level of the deep, doubly occupied F0 level is
squeezed too close to the valence band maximum, directly
resulting in a low calculated formation energy. Our finding here is similar to the findings for the formation energies of neutral, interstitial Si atoms in silicon [14–16], for
which DMC calculations show that DFT underestimates
formation energies in the case of occupied mid-gap defect
levels. This analysis is also consistent with recent QMC
results for defect formation energies in aluminum [18] for metallic systems, for which DFT delocalization problems are less significant, the DMC results are more closely
matched.
For the charged defects F+1 and F+2 , the comparison between the DFT and QMC results is more complex.
The difference between the QMC and DFT -computed
formation energies arises from an interplay between the
difference in predicted band gap, the occupation of the
Kohn-Sham defect level (zero, one, and two electrons for
the F+2 , F+1 , and F0 center respectively), and the fact
that the charged-defect interactions arising from the use
of periodic boundary conditions are likely different for
the two schemes. That is, since the degree of localization/delocalization is different, screening is also expected
to be different for the two methodologies - most likely enhanced in the case of DFT.
We now turn to the QMC description of the optical ionization energies, corresponding to vertical FranckCondon transitions on a configuration coordinate diagram as illustrated in Fig. 3. An optical transition occurs when a photon is absorbed or emitted by the defect;
because this transition essentially takes place instantaneously on the scale of lattice relaxations it occurs at
fixed atomic coordinates (and hence is represented as a
vertical transition). Such a transition places the system
in an excited vibrational state; for example, F0 -center absorption illustrated in Fig. 3 refers to the absorption of a
photon and the promotion of an electron from the filled
mid-gap level to the conduction band, leaving behind an
electron in the conduction band and an F+1 center in an
excited vibrational state (which soon decays to the F+1
vibrational ground state). Therefore, we compute the optical transitions by using the relaxed coordinates of the
initial state, and occupying the Kohn-Sham orbitals as
appropriate to describe the vibronic state.
Our DMC absorption energies (Fig 3) are in excellent
agreement with experiment and remarkably close to the
GW-computed values, demonstrating the high-accuracy
potential of the DMC methodology. The DMC emission
energies are also remarkably similar to the GW-computed
values, but in disagreement with the experimental numbers. The disagreement between the GW and experimental values led the authors in [8] to suggest that the
low energy signal around 2.3-2.4 eV that is observed in
fact arises when electrons in the defect level recombine
with holes in the valence band. We find it notable that
two distinct many-body approaches (namely QMC and
GW) have yielded similar results for the optical emission
transitions in question.
This leads us to suggest two possibilities. First, we find
it likely that, as suggested by the authors in Ref. [8], the
original emission-peak assignment should be revisited. A
second possibility is based on the fact that both our QMC
and the GW results are built from DFT-relaxed atomic
geometries (Table II). It is possible that the GW and
QMC results compare favorably because both methods
are using similar DFT-relaxed lattice geometries. If the
relaxations are not properly described in DFT, then the
many-body energies may be similar but incorrect. However, the possibility that the lattice geometries are problematic seems unlikely given the exceptional agreement
with experiment for the absorption transitions.
In conclusion, we demonstrate the application of quantum Monte Carlo methods to the calculation of the thermal and optical ionization energies of point defects in
solids. The striking agreement between two highly accurate methods, quantum Monte Carlo and GW, suggests
5
that predictive calculations of point defect properties are
now in reach. Due to its inherently many-body approach
and accurate treatment of electron correlation, quantum
Monte Carlo shows large promise for the quantitative
first-principles calculation of point defect properties.
We gratefully acknowledge fruitful discussions with
S.B. Zhang, Y.Y. Sun, P. Zhang, and T. Abtew. This
work was supported by DOE grant de-sc0002623. Calculations were performed in part at the National Energy
Research Scientific Computing Center of the Lawrence
Berkeley National Laboratory and in part by the National Science Foundation through TeraGrid resources
provided by NCSA under grant TG-DMR090027.
6
I. Properties of MgO Molecule as Described in
Diffusion Monte Carlo
We compute the properties of the MgO molecule in
diffusion Monte Carlo (DMC) using both small (helium) and large (neon) -core pseudopsotentials; the results are given in Table III. While both pseudopotentials give excellent agreement to the experimental bond
length and electron affinity, we find some difference for
the binding energy and the ionization energy. Using a
He-core, rather than Ne-core, pseudopotential for Mg increases the molecular binding energy from 2.28±0.01 eV
to 2.43±0.01 eV, in comparison to the experimental value
of 2.54±0.22 eV [19, 46–48]. This suggests that allowing
the Mg 2s and 2p electrons to participate in the bonding allows more recovery of the binding energy. We also
find that using the Ne-core pseudopotential introduces a
small 0.04(2) eV error in the ionization energy compared
to the He core pseudopotential.
bond
binding
electron
length (Å) energy (eV) affinity (eV)
DMC, Ne-core PP
DMC, Ar-core PP
Exp. (Refs. [19, 46–48])
1.75
1.75
1.75
2.28(1)
2.43(1)
2.56(21)
1.76(1))
1.72(1)
1.630(25)
TABLE III. Comparison of bond length, binding energy, and
electron affinity for the MgO molecule according to DMC and
in experiment. Two sets of DMC results are provided, corresponding to the use of neon (large) and helium (small) -core
pseudopotentials. Error bars are shown in parenthesis.
II. Properties of Crystalline MgO as Described
in Diffusion Monte Carlo
The extrapolation framework described in Refs. [40–
42] is used to compute the atomization energy for the
MgO solid, as shown in Fig. 4. We use supercells containing 16, 32, and 64 atoms using twist-averaged boundary conditions. The dependence of the binding energy
on the supercell size reflects the spurious electron correlation that appears in many-body theories when periodic boundary conditions are applied. This spurious
correlation disappears in the infinite size supercell limit.
The results presented in Fig. 4 are computed using twist–
averaged boundary conditions. The extrapolated value of
the atomization energy in DMC is 10.18±0.05 eV per formula unit, in comparison to the experimental and DFT
values of 10.5 eV and 9.48 eV, respectively [19–21]. Although DMC improves the atomization energy in comparison to DFT, it is most likely necessary to include the
Mg 2s and 2p electrons in valence to obtain atomization
energies closer to experiment.
The optical gap, ionization potential, and electron
affinity are similarly computed by extrapolation, as illus-
trated in Fig. 5. Here we use only the Γ-point orbitals to
construct the many-particle wave function, to save computational cost and since the MgO solid exhibits a direct
band gap at the Γ-point. For these quantities, additional
finite size effects are present including (1) periodic image interactions between the electron–hole pair for the
optical gap and (2) the electrostatic interaction between
charged supercells in the calculation of the ionization potential and electron affinity. Extrapolating to the infinite
supercell limit, we obtain an optical gap of 7.96±0.06 eV
and a quasiparticle gap of 7.89±0.10 eV, in close agreement with the experimental band gap of 7.8 eV.
Atomization Energy (eV/MgO)
Supplementary Information
10.5
Experiment
10.0
9.5
9.0
DFT-PBE
1/64
1/32
1/(number of atoms)
1/16
FIG. 4. (Color online). The extrapolated value of the atomization energy as computed in DMC is 10.18±0.05 eV, in
comparison to the experimental and DFT values of 10.5 eV
and 9.48 eV, respectively [19–21].
7
Energy (eV)
a)
7.5
Energy (eV)
7.0
a)
6.5
6.0
b)
Γ-point optical excitation
8.0
12
1/128 1/64
1/16
1/32
Γ-point quasiparticle gap
1/64
1/32
10
E(N+1)-E(N)
8
6
4
2
E(N)-E(N-1)
1/(number of atoms)
1/16
FIG. 5.
(Color online). Optical and quasiparticle gap
in MgO obtained by extrapolating supercells to infinite
size. The data points for finite-size supercells are computed with DMC. The extrapolated value of the optical
gap is 7.96±0.06 eV. The extrapolated ionization potential IP=E(N)-E(N-1)=3.28±0.07 eV. The extrapolated electron affinity EA=E(N+1)-E(N)=11.17±0.07 eV. This gives a
quasiparticle gap of QP=EA-IP=7.89±0.10 eV. (Note the EA
and IP are referenced to the average electrostatic potential in
the supercell.)
8
∗
†
‡
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
Currently at University of Illinois, [email protected]
Currently at University of Illinois, [email protected]
e-mail:[email protected]
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